Bourbaki

Corollary 2

Tag 07ZO, permanent. Cite it as [Bourbaki, Tag 07ZO].

Topological Vector Spaces, V, p. 23, § 2. Orthogonal families in a hilbertian space

Every hilbertian space is isomorphic to a space ℓ2(I)\ell^2(I).

More precisely, let (ei)i∈I(e_i)_{i \in I} be an orthonormal basis of a hilbertian space E. By props. 4 (V, p. 21) and 5 (V, p. 22), the mapping ϕ\phi defined by

ϕ(x)=(⟨ei∣x⟩)i∈I \phi(x) = (\langle e_i|x \rangle)_{i \in I}
is a hilbertian space isomorphism from E onto ℓ2(I)\ell^2(I). The inverse isomorphism ψ\psi is defined by
ψ((λi)i∈I)=∑i∈Iλiei. \psi((\lambda_i)_{i \in I}) = \sum_{i \in I} \lambda_i e_i .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Transcribed from the printed volume. Not translated.